DecomposeDessin - Maple Help
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GroupTheory

  

FindDessins

  

find all dessins d'enfants with a specified branch pattern

  

DecomposeDessin

  

find all decompositions of a Belyi map represented by a dessin

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

FindDessins( B0, B1, Binf )

DecomposeDessin( d, L, Gr )

Parameters

B0, B1, Binf

-

three lists of positive integers, each with the same sum n

d

-

list [ g0, g1 ] representing a conjugacy class of -constellations or, equivalently, a dessin

L

-

(optional) name

Gr

-

(optional) name

Description

• 

Let  be a positive integer. A 3-constellation of degree  is a triplet  of elements of that generate a transitive subgroup of  and satisfy .

• 

Two -constellations ,  are conjugated if there exists  in  with  for each  in  (or, equivalently, each  in ).

• 

The branch pattern of  is a triplet  where  is a partition of  giving the cycle-structure of . We include -cycles so FindDessins can find  by taking the sum of the entries of each .

• 

Given B0, B1, Binf as input, FindDessins computes one representative from every conjugacy class of -constellations with branch pattern (B0, B1, Binf). Each -constellation  will be represented by the list , since  can be computed as .

• 

A conjugacy class of -constellations corresponds 1-1 with a dessin d'enfant, as well as with a Belyi map (up to equivalence). A Belyi map is a holomorphic function from a compact Riemann surface to the Riemann sphere that only ramifies above {0,1,infinity}. So we can count how many dessins, or how many Belyi maps, exists for a given branch pattern by counting the output of FindDessins.

• 

FindDessins implements the strategy of Section  in arXiv:1604.08158 with a number of additions. Progress is reported during the computation by setting infolevel['FindDessins'] to 1 or 2.

Examples

Suppose we want to know if there exists a Belyi map  whose branch pattern above , ,  is [1$39], [2$14], [7$4]. This means that  should have  root of order  and  roots of order ,  should have  roots of order ,  should have  poles of order , and  should be unramified outside of {0,1,infinity}. We can determine if such  exist (and if so, how many) as follows.

(1)

(2)

Found  conjugacy class of -constellations (i.e.  dessin), so there exists a Belyi map (unique up to equivalence) with branch pattern B.

(3)

Now let's check that d = [ g0, g1 ] has branch pattern B.

(4)

g0 indeed has cycle-structure [1,3$9] (a -cycle and  -cycles)

(5)

g1 has cycle-structure [2$14] ( -cycles)

(6)

Has cycle structure [7$4] ( -cycles).

(7)

The Belyi map for d is indecomposable.

Example with decompositions

(8)

(9)

(10)

(11)

The Belyi map for S[1] has  decompositions. With additional arguments, DecomposeDessin returns a list with information on each Fn, and a decomposition graph.

(12)

(13)

(14)

Decomposition graph:

F1 .. F5 have the same dessin so they represent the same Belyi map (of degree = ). The reason for listing all five is because their degree = 2 decomposition factors differ.

Compatibility

• 

The GroupTheory[FindDessins] and GroupTheory[DecomposeDessin] commands were introduced in Maple 2019.

• 

For more information on Maple 2019 changes, see Updates in Maple 2019.

See Also

GroupTheory

http://oeis.org/A112948

 


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